MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  noinfbnd1lem4 Structured version   Visualization version   GIF version

Theorem noinfbnd1lem4 27226
Description: Lemma for noinfbnd1 27229. If 𝑈 is a prolongment of 𝑇 and in 𝐵, then (𝑈‘dom 𝑇) is not undefined. (Contributed by Scott Fenton, 9-Aug-2024.)
Hypothesis
Ref Expression
noinfbnd1.1 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
Assertion
Ref Expression
noinfbnd1lem4 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ ∅)
Distinct variable groups:   𝐵,𝑔,𝑢,𝑣,𝑥,𝑦   𝑣,𝑈   𝑥,𝑢,𝑦   𝑔,𝑉   𝑥,𝑣,𝑦,𝑈
Allowed substitution hints:   𝑇(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑈(𝑢,𝑔)   𝑉(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem noinfbnd1lem4
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 simpl1 1191 . . . . . . . 8 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → ¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
2 simpl2 1192 . . . . . . . 8 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝐵 No 𝐵𝑉))
3 simprl 769 . . . . . . . 8 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → 𝑤𝐵)
4 simpl3 1193 . . . . . . . . 9 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇))
5 simp2l 1199 . . . . . . . . . . . . 13 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → 𝐵 No )
65sselda 3982 . . . . . . . . . . . 12 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ 𝑤𝐵) → 𝑤 No )
7 simp3l 1201 . . . . . . . . . . . . . 14 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → 𝑈𝐵)
85, 7sseldd 3983 . . . . . . . . . . . . 13 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → 𝑈 No )
98adantr 481 . . . . . . . . . . . 12 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ 𝑤𝐵) → 𝑈 No )
10 sltso 27176 . . . . . . . . . . . . 13 <s Or No
11 soasym 5619 . . . . . . . . . . . . 13 (( <s Or No ∧ (𝑤 No 𝑈 No )) → (𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤))
1210, 11mpan 688 . . . . . . . . . . . 12 ((𝑤 No 𝑈 No ) → (𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤))
136, 9, 12syl2anc 584 . . . . . . . . . . 11 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ 𝑤𝐵) → (𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤))
1413impr 455 . . . . . . . . . 10 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → ¬ 𝑈 <s 𝑤)
153, 14jca 512 . . . . . . . . 9 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑤𝐵 ∧ ¬ 𝑈 <s 𝑤))
16 noinfbnd1.1 . . . . . . . . . 10 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
1716noinfbnd1lem2 27224 . . . . . . . . 9 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑤𝐵 ∧ ¬ 𝑈 <s 𝑤))) → (𝑤 ↾ dom 𝑇) = 𝑇)
181, 2, 4, 15, 17syl112anc 1374 . . . . . . . 8 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑤 ↾ dom 𝑇) = 𝑇)
1916noinfbnd1lem3 27225 . . . . . . . 8 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑤𝐵 ∧ (𝑤 ↾ dom 𝑇) = 𝑇)) → (𝑤‘dom 𝑇) ≠ 1o)
201, 2, 3, 18, 19syl112anc 1374 . . . . . . 7 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑤‘dom 𝑇) ≠ 1o)
2120neneqd 2945 . . . . . 6 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑤𝐵𝑤 <s 𝑈)) → ¬ (𝑤‘dom 𝑇) = 1o)
2221expr 457 . . . . 5 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ 𝑤𝐵) → (𝑤 <s 𝑈 → ¬ (𝑤‘dom 𝑇) = 1o))
23 imnan 400 . . . . 5 ((𝑤 <s 𝑈 → ¬ (𝑤‘dom 𝑇) = 1o) ↔ ¬ (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1o))
2422, 23sylib 217 . . . 4 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ 𝑤𝐵) → ¬ (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1o))
2524nrexdv 3149 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → ¬ ∃𝑤𝐵 (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1o))
26 breq2 5152 . . . . . . 7 (𝑥 = 𝑈 → (𝑦 <s 𝑥𝑦 <s 𝑈))
2726rexbidv 3178 . . . . . 6 (𝑥 = 𝑈 → (∃𝑦𝐵 𝑦 <s 𝑥 ↔ ∃𝑦𝐵 𝑦 <s 𝑈))
28 simpl1 1191 . . . . . . 7 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → ¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
29 dfral2 3099 . . . . . . . 8 (∀𝑥𝐵𝑦𝐵 𝑦 <s 𝑥 ↔ ¬ ∃𝑥𝐵 ¬ ∃𝑦𝐵 𝑦 <s 𝑥)
30 ralnex 3072 . . . . . . . . 9 (∀𝑦𝐵 ¬ 𝑦 <s 𝑥 ↔ ¬ ∃𝑦𝐵 𝑦 <s 𝑥)
3130rexbii 3094 . . . . . . . 8 (∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ↔ ∃𝑥𝐵 ¬ ∃𝑦𝐵 𝑦 <s 𝑥)
3229, 31xchbinxr 334 . . . . . . 7 (∀𝑥𝐵𝑦𝐵 𝑦 <s 𝑥 ↔ ¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
3328, 32sylibr 233 . . . . . 6 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → ∀𝑥𝐵𝑦𝐵 𝑦 <s 𝑥)
34 simpl3l 1228 . . . . . 6 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → 𝑈𝐵)
3527, 33, 34rspcdva 3613 . . . . 5 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → ∃𝑦𝐵 𝑦 <s 𝑈)
36 breq1 5151 . . . . . 6 (𝑦 = 𝑤 → (𝑦 <s 𝑈𝑤 <s 𝑈))
3736cbvrexvw 3235 . . . . 5 (∃𝑦𝐵 𝑦 <s 𝑈 ↔ ∃𝑤𝐵 𝑤 <s 𝑈)
3835, 37sylib 217 . . . 4 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → ∃𝑤𝐵 𝑤 <s 𝑈)
39 simpl2l 1226 . . . . . . . . . 10 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → 𝐵 No )
4039adantr 481 . . . . . . . . 9 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → 𝐵 No )
41 simprl 769 . . . . . . . . 9 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → 𝑤𝐵)
4240, 41sseldd 3983 . . . . . . . 8 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → 𝑤 No )
4334adantr 481 . . . . . . . . 9 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → 𝑈𝐵)
4440, 43sseldd 3983 . . . . . . . 8 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → 𝑈 No )
45 simpl2 1192 . . . . . . . . . . 11 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → (𝐵 No 𝐵𝑉))
4616noinfno 27218 . . . . . . . . . . 11 ((𝐵 No 𝐵𝑉) → 𝑇 No )
4745, 46syl 17 . . . . . . . . . 10 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → 𝑇 No )
4847adantr 481 . . . . . . . . 9 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → 𝑇 No )
49 nodmon 27150 . . . . . . . . 9 (𝑇 No → dom 𝑇 ∈ On)
5048, 49syl 17 . . . . . . . 8 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → dom 𝑇 ∈ On)
51 simpll1 1212 . . . . . . . . . 10 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → ¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥)
52 simpll2 1213 . . . . . . . . . 10 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝐵 No 𝐵𝑉))
53 simpll3 1214 . . . . . . . . . 10 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇))
54 simprr 771 . . . . . . . . . . . 12 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → 𝑤 <s 𝑈)
5542, 44, 12syl2anc 584 . . . . . . . . . . . 12 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤))
5654, 55mpd 15 . . . . . . . . . . 11 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → ¬ 𝑈 <s 𝑤)
5741, 56jca 512 . . . . . . . . . 10 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑤𝐵 ∧ ¬ 𝑈 <s 𝑤))
5851, 52, 53, 57, 17syl112anc 1374 . . . . . . . . 9 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑤 ↾ dom 𝑇) = 𝑇)
59 simpl3r 1229 . . . . . . . . . 10 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → (𝑈 ↾ dom 𝑇) = 𝑇)
6059adantr 481 . . . . . . . . 9 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑈 ↾ dom 𝑇) = 𝑇)
6158, 60eqtr4d 2775 . . . . . . . 8 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑤 ↾ dom 𝑇) = (𝑈 ↾ dom 𝑇))
62 simplr 767 . . . . . . . 8 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑈‘dom 𝑇) = ∅)
63 nogt01o 27196 . . . . . . . 8 (((𝑤 No 𝑈 No ∧ dom 𝑇 ∈ On) ∧ ((𝑤 ↾ dom 𝑇) = (𝑈 ↾ dom 𝑇) ∧ 𝑤 <s 𝑈) ∧ (𝑈‘dom 𝑇) = ∅) → (𝑤‘dom 𝑇) = 1o)
6442, 44, 50, 61, 54, 62, 63syl321anc 1392 . . . . . . 7 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ (𝑤𝐵𝑤 <s 𝑈)) → (𝑤‘dom 𝑇) = 1o)
6564expr 457 . . . . . 6 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ 𝑤𝐵) → (𝑤 <s 𝑈 → (𝑤‘dom 𝑇) = 1o))
6665ancld 551 . . . . 5 ((((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) ∧ 𝑤𝐵) → (𝑤 <s 𝑈 → (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1o)))
6766reximdva 3168 . . . 4 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → (∃𝑤𝐵 𝑤 <s 𝑈 → ∃𝑤𝐵 (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1o)))
6838, 67mpd 15 . . 3 (((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) ∧ (𝑈‘dom 𝑇) = ∅) → ∃𝑤𝐵 (𝑤 <s 𝑈 ∧ (𝑤‘dom 𝑇) = 1o))
6925, 68mtand 814 . 2 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → ¬ (𝑈‘dom 𝑇) = ∅)
7069neqned 2947 1 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ (𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2709  wne 2940  wral 3061  wrex 3070  cun 3946  wss 3948  c0 4322  ifcif 4528  {csn 4628  cop 4634   class class class wbr 5148  cmpt 5231   Or wor 5587  dom cdm 5676  cres 5678  Oncon0 6364  suc csuc 6366  cio 6493  cfv 6543  crio 7363  1oc1o 8458   No csur 27140   <s cslt 27141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-1o 8465  df-2o 8466  df-no 27143  df-slt 27144  df-bday 27145
This theorem is referenced by:  noinfbnd1lem5  27227  noinfbnd1lem6  27228
  Copyright terms: Public domain W3C validator